Topological Tenfold Classification and the Entanglement of Two Qubits

• URL: http://arxiv.org/abs/2502.02427v1
• Date: Tue, 04 Feb 2025 15:47:19 GMT
• Title: Topological Tenfold Classification and the Entanglement of Two Qubits
• Authors: Nadav Orion, Eric Akkermans,
• Abstract summary: We use the Cartan decomposition to characterize topological properties and their connection to quantum entanglement. Topological features are obtained systematically by a mapping to a quantum graph. An additional perspective is provided regarding the extension of this new approach to condensed matter systems.
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• Abstract: We present a constructive method utilizing the Cartan decomposition to characterize topological properties and their connection to two-qubit quantum entanglement, in the framework of the tenfold classification and Wootters' concurrence. This relationship is comprehensively established for the 2-qubit system through the antiunitary time reversal (TR) operator. The TR operator is shown to identify concurrence and differentiate between entangling and non-entangling operators. This distinction is of a topological nature, as the inclusion or exclusion of certain operators alters topological characteristics. Proofs are presented which demonstrate that the 2-qubit system can be described in the framework of the tenfold classification, unveiling aspects of the connection between entanglement and a geometrical phase. Topological features are obtained systematically by a mapping to a quantum graph, allowing for a direct computation of topological integers and of the 2-qubit equivalent of topological zero-modes. An additional perspective is provided regarding the extension of this new approach to condensed matter systems, illustrated through examples involving indistinguishable fermions and arrays of quantum dots.

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